Remark 2 
From (REF ) it can be seen that, if the system input is not a function of the system parameter of interest, then \({{\partial {\bf {g}}} \mathord {\left\bad.{\vphantom{{\partial {\bf {g}}} {\partial {\bf {\gamma }}}}} \right.\hspace{0.0pt}} {\partial {\gamma }}} = \bf 0\)  and hence by the Lebesgue dominated convergence Theorem [1]}, and our assumption about the boundedness of score functions in REF , the first term of this equation will vanish,
\(\Delta I\left( {{\textbf {x}};{\textbf {y}}} \right) = {\mathop {\rm E}\nolimits } \left\lbrace  {\nabla _{{\textbf {y}}}^T \ln p_{{\textbf {x}}|{\textbf {y}}} \left( {{{\textbf {x}}}|{{\textbf {y}}}} \right) \frac{{\partial {\bf {f}}}}{{\partial {\gamma }}}{\varepsilon }_{\gamma } } \right\rbrace  + o\left( {\varepsilon }_{\gamma } \right).\) 

